as well (use the inverse permutation). Also, ifC is equivalent to D,andDis equivalent toE,then C is equivalent toE.All codes equivalent to some code C form anequivalence class. The number of all permutations of ann-set is n! The groupSn of all permutations onnobjects is thesymmetric group.
Such a clearly defined notion of equivalence is really helpful. As an exam- ple, if I want to decide the existence of a linear code with certain parameters [n, k, d]q, I can assume the generator matrix starts with a (k, k)-unit matrix, that is,G= (I|A) whereAis a (k, n−k)-matrix. Why is that so? Assume a code with these parameters exists, with generator matrix G.There must be somekcolumns ofG,which are linearly independent.
Choose a permutationσmapping thesekcoordinates into the first kcoordi- nates. The image of my code underσ is an equivalent code with a generator matrix whose k first columns are linearly independent. After Gauß elimina- tion, we find a generator matrix of the form G= (I|A).
It should now be clear when two codes are equivalent under permutations of coordinates. A highly interesting situation occurs if we find a permutation σ such thatσ(C) =C (in words: the image of our code under σ is the code itself). In this case we call σ an automorphism ofC. The automorphisms ofC form a subgroup, thestabilizeror automorphism group of C.We saw a large example in Chapter 7. The generator matrix for the binary Golay code [23,12,7]2 given there has the special feature that all rows are cyclic shifts of the first row. In other words, the cyclic permutation
σ: 17→2, 27→3, . . .227→23, 237→1 is an automorphism ofG23.
It is clear that the identity permutation (fixing all coordinates) always is an automorphism. If σ is an automorphism, then the inverse permutationσ−1 is an automorphism. Also, if σ1 and σ2 both are automorphisms, then the concatenationσ2◦σ1 is an automorphism as well. In other words, the set of all automorphisms of a codeCis a group, the automorphism group.
12.18 Definition. The permutation automorphism group of a code C of lengthnis the group of all permutationsσ of the coordinates which satisfy σ(C) =C.
As an example, consider the following generator matrix of an octal (8-ary) four-dimensional code of length 16 :
1 0 0 0a1a2a3a4a21a22a23a24a41 a42 a43a44 0 1 0 0a2a1a4a3a22a21a24a23a42 a41 a44a43 0 0 1 0a3a4a1a2a23a24a21a22a43 a44 a41a42 0 0 0 1a4a3a2a1a24a23a22a21a44 a43 a42a41
Here (a1, a2, a3, a4) = (ǫ6, ǫ2, ǫ,1) and the same representation of F8 as in Chapter 4 is used. The parameters of this code are [16,4,12]8,but we do not want to prove this here.
Some automorphisms are clearly visible. The permutation σ1: 1↔2, 3↔4, . . .15↔16
maps the first row to the second and back, the third row to the fourth and back. Similarly,
σ2: 1↔3, 2↔4, . . .13↔15, 14↔16
is an automorphism. It becomes obvious that our notation for permutations is rather clumsy. Here is a better notation:
σ1= (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16) σ2= (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)
The rule is: inside each parenthesis we map to the right neighbor, and the end of the parenthesis is mapped back to the beginning. Each permutation can be written in this cycle notation. The concatenationσ2◦σ1 (apply at firstσ1,thenσ2) is also an automorphism, in cycle notation
σ2◦σ1= (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,16)(14,15).
The◦ can also be omitted. It is easy to check thatσ2σ1=σ1σ2.These four automorphisms, the identity permutation, σ1, σ2 and σ2σ1 form a group, a subgroup of the automorphism group.
Monomial equivalence in the linear case
There is a more general notion of equivalence for q-ary linear codes. We call it monomial equivalencein the linear case. The group of motions is larger: we are allowed to multiply by nonzero elements of the field in each coordinate, and we permute the coordinates. The corresponding group has order (number of elements) (q−1)nn!
Each such mapping respects most of the basic properties of codes: length, number of codewords, distance distribution, strength.
12.19 Definition. The monomial automorphism group (or simply the automorphism group) of a linear code C of length n is the group of all monomial operationsσ of the coordinates which mapC toC.
So why is this larger group not always used; in other words, why is the restricted notion of equivalence based on permutations of coordinates alone so popular? Probably there are two reasons. One is that permutations of coordinates are easier to work with. General monomial operations are much
harder to see. A more serious reason is that there are important invariants of linear codes which are respected by permutations of coordinates but not by monomial operations. The most important are the dimensions of trace codes and subfield codes with respect to some subfield. It is clear that permutations of coordinates do respect those, and it is easy to find examples which show that general monomial operations do not. In other words, it can happen that the subfield code of a code and the subfield code of a monomial image of the code have different dimensions. Observe that this is a problem only if q is not a prime. Also, in the binary case (linear) monomial equivalence and permutation equivalence coincide. We consider monomial equivalence and the automorphism group of Definition 12.19 the most natural notion in the case of linear codes.
General monomial equivalence
An even more general notion of monomial equivalence can be used when linearity is not an issue.
We allow an arbitrary permutation of the alphabet in each coordinate as well as an arbitrary permutation of the coordinates. The corresponding group has order (q!)nn! Clearly, it can happen that these general monomial opera- tions map linear codes (ifqis a prime-power) to nonlinear codes.
Cyclic codes
A code of lengthn iscyclicif it admits an automorphism in the group of permutations of coordinates which consists of one single n-cycle. By passing to an equivalent code it can be assumed that this automorphism is
σ = (1,2, . . . , n). Normally notation is chosen such that this is the case. In the next chapter we develop the theory of linear cyclic codes whenn and q are coprime. We start from a different point of view. It will be easy to see that the codes we describe are cyclic. The fact that the inverse is true is not too hard to prove either. Each cyclic linear code satisfying gcd(q, n) = 1 is one of the codes described in the following chapter.
1. Theautomorphism group of a code of lengthnis its stabilizer in thesymmetric groupSn,whereSn consists of all permutations of coordinates.
2. Two codes areequivalent(orisomorphic) if one can be obtained from the other by an application of an element ofSn. 3. The basic coding parameters are preserved by the larger lin-
ear monomial group (permute coordinates and multiply by arbitrary nonzero field elements in each coordinate).
4. Warning: if two codes are linearly monomially equivalent, their subfield codes can be much different.
The same is true of their trace codes.
5. A linear code of lengthnis cyclic if it has a cyclic permutation of coordinates consisting of one cycle of lengthnas an automorphism.
Exercises 12.4
12.4.1. Find an example of two linear quaternary codes, which are
monomially equivalent but have subfield codes and trace codes (down toF2) of different dimension.
12.4.2. Find all cyclic binary codes of length 3 (with respect to the cyclic permutation σ= (1,2,3)).
12.4.3. Show that the rule σ2σ1=σ1σ2 isnottrue in general for permutationsσ1, σ2.
12.4.4. Can a permutation on nobjects have order larger thann?
12.4.5. Determine the automorphism group of the extended binary Hamming code [8,4,4]2,in particular, its order and its degree of transitivity.
12.4.6. Consider the extended binary Hamming code[8,4,4]2. Count the unordered 4-sets of coordinates with the property that the corresponding columns of a generator matrix sum to 0.
Prove that they form a Steiner system S(3,4,8).